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add chapter about diagonal and permutation matrices to manual
author Jaroslav Hajek <highegg@gmail.com>
date Mon, 23 Feb 2009 13:55:44 +0100
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8839
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1 @c Copyright (C) 2009 Jaroslav Hajek
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2 @c
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3 @c This file is part of Octave.
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4 @c
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5 @c Octave is free software; you can redistribute it and/or modify it
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6 @c under the terms of the GNU General Public License as published by the
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7 @c Free Software Foundation; either version 3 of the License, or (at
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8 @c your option) any later version.
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9 @c
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10 @c Octave is distributed in the hope that it will be useful, but WITHOUT
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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13 @c for more details.
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14 @c
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15 @c You should have received a copy of the GNU General Public License
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16 @c along with Octave; see the file COPYING. If not, see
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17 @c <http://www.gnu.org/licenses/>.
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18
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19 @ifhtml
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20 @set htmltex
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21 @end ifhtml
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22 @iftex
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23 @set htmltex
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24 @end iftex
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25
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26 @node Diagonal and Permutation Matrices
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27 @chapter Diagonal and Permutation Matrices
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28
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29 @menu
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30 * Basic Usage:: Creation and Manipulation of Diagonal and Permutation Matrices
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31 * Matrix Algebra:: Linear Algebra with Diagonal and Permutation Matrices
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32 * Function Support:: Functions That Are Aware of These Matrices
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33 * Example Codes:: Some Examples of Usage
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34 * Zeros Treatment:: The Differences in Treatment of Zero Elements
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35 @end menu
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36
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37 @node Basic Usage
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38 @section The Creation and Manipulation of Diagonal and Permutation Matrices
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39
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40 A diagonal matrix is defined as a matrix that has zero entries outside the main diagonal;
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41 that is, @code{D(i,j) == 0} if @code{i != j}.
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42 Most often, square diagonal matrices are considered; however, the definition can equally
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43 be applied to nonsquare matrices, in which case we usually speak of a rectangular diagonal
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44 matrix. For more information, see @url{http://en.wikipedia.org/wiki/Diagonal_matrix}.
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45
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46 A permutation matrix is defined as a square matrix that has a single element equal to unity
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47 in each row and each column; all other elements are zero. That is, there exists a
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48 permutation (vector) @code{p} such that @code{P(i,j) == 1} if @code{j == p(i)} and
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49 @code{P(i,j) == 0} otherwise. For more information, see
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50 @url{http://en.wikipedia.org/wiki/Permutation_matrix}.
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51
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52 Octave provides special treatment of real and complex rectangular diagonal matrices,
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53 as well as permutation matrices. They are stored as special objects, using efficient
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54 storage and algorithms, facilitating writing both readable and efficient matrix algebra
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55 expressions in the Octave language.
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56
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57 @menu
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58 * Creating Diagonal Matrices::
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59 * Creating Permutation Matrices::
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60 * Explicit and Implicit Conversions::
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61 @end menu
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62
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63 @node Creating Diagonal Matrices
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64 @subsection Creating Diagonal Matrices
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65
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66 The most common and easiest way to create a diagonal matrix is using the built-in
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67 function @dfn{diag}. The expression @code{diag (@var{v})}, with @var{v} a vector,
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68 will create a square diagonal matrix with elements on the main diagonal given
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69 by the elements of @var{v}, and size equal to the length of @var{v}.
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70 @code{diag (@var{v}, m, n)} can be used to construct a rectangular diagonal matrix.
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71 The result of these expressions will be a special diagonal matrix object, rather
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72 than a general matrix object.
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73
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74 Some other built-in functions can also return diagonal matrices. Examples include
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75 @dfn{balance} or @dfn{inv}.
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76
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77 @DOCSTRING(diag)
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78
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79 @node Creating Permutation Matrices
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80 @subsection Creating Permutation Matrices
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81
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82 For creating permutation matrices, Octave does not introduce a new function, but
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83 rather overrides an existing syntax. That is, if @var{q} is a permutation vector
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84 of length @var{n}, the expression
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85 @example
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86 @var{P} = eye (@var{n}) (:, @var{q});
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87 @end example
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88 will create a permutation matrix - a special matrix object.
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89 @example
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90 eye (@var{n}) (@var{q}, :)
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91 @end example
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92 will also work (a row permutation matrix), as well as
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93 @example
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94 eye (@var{n}) (@var{q1}, @var{q2}).
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95 @end example
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96 Note that the identity, @code{eye (@var{n})}, is a diagonal matrix by definition,
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97 but should work in any place where a permutation matrix is requested.
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98
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99 Some other built-in functions can also return permutation matrices. Examples include
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100 @dfn{inv} or @dfn{lu}.
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101
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102 @node Explicit and Implicit Conversions
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103 @subsection Explicit and Implicit Conversions
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104
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105 The diagonal and permutation matrices are special objects in their own right. A number
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106 of operations and built-in functions, are defined for these matrices to use special,
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107 more efficient code than would be used for a full matrix in the same place. Examples
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108 are given in further sections.
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109
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110 To facilitate smooth mixing with full matrices, backward compatibility, and
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111 compatibility with Matlab, the diagonal and permutation matrices should allow
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112 any operation that works on full matrices, and will either treat is specially,
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113 or implicitly convert themselves to full matrices.
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114
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115 Instances include matrix indexing (except for extracting a single element),
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116 indexed assignment, or applying most mapper functions, such as @dfn{exp}.
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117
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118 An explicit conversion to a full matrix can be requested using the built-in
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119 function @dfn{full}. It should also be noted that the diagonal and permutation
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120 matrix objects will cache the result of the conversion after it is first
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121 requested (explicitly or implicitly), so that subsequent conversions will
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122 be very cheap.
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123
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124 @node Matrix Algebra
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125 @section Linear Algebra with Diagonal and Permutation Matrices
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126
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127 As has been already said, diagonal and permutation matrices make it
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128 possible to use efficient algorithms while preserving natural linear
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129 algebra syntax. This section describes in detail the operations that
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130 are treated specially when performed on these special matrix objects.
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131
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132 @menu
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133 * Expressions Involving Diagonal Matrices::
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134 * Expressions Involving Permutation Matrices::
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135 @end menu
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136
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137 @node Expressions Involving Diagonal Matrices
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138 @subsection Expressions Involving Diagonal Matrices
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139
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140 Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix,
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141 then @code{@var{D}*@var{M}} will scale the rows of @var{M}. That means,
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142 if @code{@var{S} = @var{D}*@var{M}}, then for each pair of indices
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143 i,j it holds
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144 @example
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145 S(i,j) = D(i,i) * M(i,j).
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146 @end example
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147 Similarly, @code{M*D} will do a column scaling.
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148
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149 The matrix @var{D} may also be rectangular, m-by-n where @code{m != n}.
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150 If @code{m < n}, then the expression @code{D*M} is equivalent to
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151 @example
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152 D(:,1:m) * M(1:m,:),
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153 @end example
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154 i.e. trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n},
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155 then @code{D*M} is equivalent to
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156 @example
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157 [D(1:n,n) * M; zeros(m-n, columns (M))],
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158 @end example
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159 i.e. null rows are appended to the result.
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160 The situation for right-multiplication @code{M*D} is analogous.
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161
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162 The expressions @code{D \ M} and @code{M / D} perform inverse scaling.
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163 They are equivalent to solving a diagonal (or rectangular diagonal)
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164 in a least-squares minimum-norm sense. In exact arithmetics, this is
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165 equivalent to multiplying by a pseudoinverse. The pseudoinverse of
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166 a rectangular diagonal matrix is again a rectangular diagonal matrix
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167 of the same dimensions, where each nonzero diagonal element is replaced
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168 by its reciprocal.
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169 The matrix division algorithms do, in fact, use division rather than
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170 multiplication by reciprocals for better numerical accuracy; otherwise, they
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171 honor the above definition. Note that a diagonal matrix is never truncated due
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172 to ill-conditioning; otherwise, it would not be much useful for scaling. This
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173 is typically consistent with linear algebra needs. A full matrix that only
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174 happens to be diagonal (an is thus not a special object) is of course treated
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175 normally.
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176
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177 If @var{D1} and @var{D2} are both diagonal matrices, then the expressions
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178 @example
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179 @var{D1} + @var{D2}
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180 @var{D1} - @var{D2}
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181 @var{D1} * @var{D2}
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182 @var{D1} / @var{D2}
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183 @var{D1} \ @var{D2}
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184 @end example
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185 again produce diagonal matrices, provided that normal
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186 dimension matching rules are obeyed. The relations used are same as described above.
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187
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188 Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or raised
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189 to a scalar power if it is square, producing diagonal matrix result in all cases.
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190
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191 A diagonal matrix can also be transposed or conjugate-transposed, giving the expected
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192 result. Extracting a leading submatrix of a diagonal matrix, i.e. @code{@var{D}(1:m,1:n)},
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193 will produce a diagonal matrix, other indexing expressions will implicitly convert to
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194 full matrix.
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195
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196 When involved in expressions with the element-by-element operators @code{.*},
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197 @code{./}, @code{.\} or @code{.^}, an implicit conversion to full matrix will
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198 also take place. This is not always strictly necessary but chosen to facilitate
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199 better consistency with Matlab.
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200
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201 @node Expressions Involving Permutation Matrices
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202 @subsection Expressions Involving Permutation Matrices
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203
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204 If @var{P} is a permutation matrix and @var{M} a matrix, the expression
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205 @code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will
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206 yield a column permutation.
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207 Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation.
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208
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209 The previously described syntax for creating permutation matrices can actually
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210 help an user to understand the connection between a permutation matrix and
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211 a permuting vector. Namely, the following holds, where @code{I = eye (n)}
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212 is an identity matrix:
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213 @example
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214 I(p,:) * M = (I*M) (p,:) = M(p,:)
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215 @end example
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216 Similarly,
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217 @example
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218 M * I(:,p) = (M*I) (:,p) = M(:,p)
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219 @end example
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220
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221 The expressions @code{I(p,:)} and @code{I(:,p)} are permutation matrices.
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222
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223 A permutation matrix can be transposed (or conjugate-transposed, which is the
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224 same, because a permutation matrix is never complex), inverting the
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225 permutation, or equivalently, turning a row-permutation matrix into a
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226 column-permutation one. For permutation matrices, transpose is equivalent to
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227 inversion, thus @code{P\M} is equivalent to @code{P'*M}. Transpose of a
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228 permutation matrix (or inverse) is a constant-time operation, flipping only a
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229 flag internally, and thus the choice between the two above equivalent
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230 expressions for inverse permuting is completely up to the user's taste.
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231
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232 Two permutation matrices can be multiplied or divided (if their sizes match), performing
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233 a composition of permutations. Also a permutation matrix can be indexed by a permutation
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234 vector (or two vectors), giving again a permutation matrix.
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235 Any other operations do not generally yield a permutation matrix and will thus
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236 trigger the implicit conversion.
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237
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238 @node Function Support
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239 @section Functions That Are Aware of These Matrices
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240
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241 This section lists the built-in functions that are aware of diagonal and
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242 permutation matrices on input, or can return them as output. Passed to other
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243 functions, these matrices will in general trigger an implicit conversion.
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244 (Of course, user-defined dynamically linked functions may also work with
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245 diagonal or permutation matrices).
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246
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247 @menu
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248 * Diagonal Matrix Functions::
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249 * Permutation Matrix Functions::
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250 @end menu
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251
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252 @node Diagonal Matrix Functions
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253 @subsection Diagonal Matrix Functions
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254
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255 @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again
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256 a diagonal matrix. @dfn{det} will use an efficient straightforward calculation
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257 when given a diagonal matrix, as well as @dfn{cond}.
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258 The following mapper functions can be applied to a diagonal matrix
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259 without converting it to a full one:
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260 @dfn{abs}, @dfn{real}, @dfn{imag}, @dfn{conj}, @dfn{sqrt}.
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261 A diagonal matrix can also be returned from the @dfn{balance}
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262 and @dfn{svd} functions.
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263
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264 @node Permutation Matrix Functions
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265 @subsection Permutation Matrix Functions
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266
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267 @dfn{inv} and @dfn{pinv} will invert a permutation matrix, preserving its
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268 specialness. @dfn{det} can be applied to a permutation matrix, efficiently
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269 calculating the sign of the permutation (which is equal to the determinant).
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270
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271 A permutation matrix can also be returned from the built-in functions
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272 @dfn{lu} and @dfn{qr}, if a pivoted factorization is requested.
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273
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274 @node Example Codes
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275 @section Some Examples of Usage
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276
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277 The following can be used to solve a linear system @code{A*x = b}
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278 using the pivoted LU factorization:
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279 @example
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280 [L, U, P] = lu (A); ## now L*U = P*A
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281 x = U \ L \ P*b;
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282 @end example
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283
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284 This is how you normalize columns of a matrix @var{X} to unit norm:
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285 @example
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286 s = norm (X, "columns");
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287 X = diag (s) \ X;
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288 @end example
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289
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290 The following expression is a way to efficiently calculate the sign of a
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291 permutation, given by a permutation vector @var{p}. It will also work
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292 in earlier versions of Octave, but slowly.
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293 @example
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294 det (eye (length (p))(p, :))
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295 @end example
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296
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297 Finally, here's how you solve a linear system @code{A*x = b}
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298 with Tikhonov regularization using SVD (a skeleton only):
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299 @example
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300 m = rows (A); n = columns (A);
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301 [U, S, V] = svd (A);
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302 ## determine the regularization factor alpha
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303 ## ...
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304 ## and solve
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305 x = U'*b;
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306 x = (S + alpha*eye (m, n)) \ x; ## this will be very efficient
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307 x = V*x;
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308 @end example
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309
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310 @node Zeros Treatment
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311 @section The Differences in Treatment of Zero Elements
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312
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313 Making diagonal and permutation matrices special matrix objects in their own
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314 right and the consequent usage of smarter algorithms for certain operations
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315 implies, as a side effect, small differences in treating zeros.
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316 The contents of this section applies also to sparse matrices, discussed in
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317 the following chapter.
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318
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319 The IEEE standard defines the result of the expressions @code{0*Inf} and
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320 @code{0*NaN} as @code{NaN}, as it has been generally agreed that this is the
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321 best compromise.
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322 Numerical software dealing with structured and sparse matrices (including
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323 Octave) however, almost always makes a distinction between a "numerical zero"
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324 and an "assumed zero".
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325 A "numerical zero" is a zero value occuring in a place where any floating-point
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326 value could occur. It is normally stored somewhere in memory as an explicit
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327 value.
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328 An "assumed zero", on the contrary, is a zero matrix element implied by the
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329 matrix structure (diagonal, triangular) or a sparsity pattern; its value is
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330 usually not stored explicitly anywhere, but is implied by the underlying
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331 data structure.
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332
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333 The primary distinction is that an assumed zero, when multiplied or divided
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334 by any number, yields *always* a zero, even when, e.g., multiplied by @code{Inf}
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335 or divided by @code{NaN}.
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336 The reason for this behavior is that the numerical multiplication is not
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337 actually performed anywhere by the underlying algorithm; the result is
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338 just assumed to be zero. Equivalently, one can say that the part of the
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339 computation involving assumed zeros is performed symbolically, not numerically.
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340
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341 This behavior not only facilitates the most straightforward and efficient
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342 implementation of algorithms, but also preserves certain useful invariants,
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343 like:
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344 @itemize
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345 @item scalar * diagonal matrix is a diagonal matrix
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346 @item sparse matrix / scalar preserves the sparsity pattern
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347 @item permutation matrix * matrix is equivalent to permuting rows
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348 @end itemize
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349 all of these natural mathematical truths would be invalidated by treating
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350 assumed zeros as numerical ones.
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351
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352 Note that certain competing software does not strictly follow this principle
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353 and converts assumed zeros to numerical zeros in certain cases, while not doing
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354 so in other cases. As of today, there are no intentions to mimick such behavior
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355 in Octave.
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356
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357 Examples of effects of assumed zeros vs. numerical zeros:
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358 @example
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359 Inf * eye (3)
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360 @result{}
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361 Inf 0 0
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362 0 Inf 0
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363 0 0 Inf
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364
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365 Inf * speye (3)
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366 @result{}
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367 Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%])
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368
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369 (1, 1) -> Inf
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370 (2, 2) -> Inf
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371 (3, 3) -> Inf
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372
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373 Inf * full (eye (3))
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374 @result{}
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375 Inf NaN NaN
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376 NaN Inf NaN
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377 NaN NaN Inf
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378
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379 @end example
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380
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381 @example
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382 diag(1:3) * [NaN; 1; 1]
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383 @result{}
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384 NaN
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385 2
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386 3
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387
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388 sparse(1:3,1:3,1:3) * [NaN; 1; 1]
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389 @result{}
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390 NaN
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391 2
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392 3
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393 [1,0,0;0,2,0;0,0,3] * [NaN; 1; 1]
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394 @result{}
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395 NaN
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396 NaN
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397 NaN
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398 @end example
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399